Difference between revisions of "User:Tohline/SSC/SoundWaves"

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A standard technique that is used throughout astrophysics to test the stability of self-gravitating fluids involves ''perturbing'' then ''linearizing'' each of the principal governing equations before seeking time-dependent solutions that simultaneously satisfy all of the equations.  When the effects of the fluid's self gravity are ignored and this analysis technique is applied to an initially homogeneous medium, the combined set of ''linearized'' governing equations generates a [http://en.wikipedia.org/wiki/Wave_equation wave equation] — whose ''general'' properties are well documented throughout the mathematics and physical sciences literature — that, specifically in our case, governs the propagation of sound waves.  It is quite advantageous, therefore, to examine how the wave equation is derived in the context of an analysis of sound waves before applying the standard perturbation/linearization technique to inhomogeneous and self-gravitating fluids.
A standard technique that is used throughout astrophysics to test the stability of self-gravitating fluids involves ''perturbing'' physical variables away from their initial (usually equilibrium) values then ''linearizing'' each of the principal governing equations before seeking time-dependent behavior of the variables that simultaneously satisfies all of the equations.  When the effects of the fluid's self gravity are ignored and this analysis technique is applied to an initially homogeneous medium, the combined set of ''linearized'' governing equations generates a [http://en.wikipedia.org/wiki/Wave_equation wave equation] — whose ''general'' properties are well documented throughout the mathematics and physical sciences literature — that, specifically in our case, governs the propagation of sound waves.  It is quite advantageous, therefore, to examine how the wave equation is derived in the context of an analysis of sound waves before applying the standard perturbation & linearization technique to inhomogeneous and self-gravitating fluids.


The discussion of sound waves provided in Chapter VIII of [[User:Tohline/Appendix/References#LL75|Landau & Lifshitz (1975)]] remains one of the best, so we will borrow heavily from it.
In what follows, we borrow heavily from Chapter VIII of [[User:Tohline/Appendix/References#LL75|Landau & Lifshitz (1975)]], as it provides an excellent introductory discussion of sound waves.


==Assembling the Key Relations==
==Assembling the Key Relations==


===Governing Equations and Supplemental Relations===
===Governing Equations and Supplemental Relations===
We begin with the set of [[User:Tohline/PGE#Principal_Governing_Equations|principal governing equations]] that provides the foundation for all of our discussions in this H_Book, except, because we are ignoring the effects of self gravity, <math>~\nabla\Phi</math> is set to zero in the Euler equation and we ignore the Poisson equation altogether.  The set of governing equations is, therefore, the
We begin with the set of [[User:Tohline/PGE#Principal_Governing_Equations|principal governing equations]] that provides the foundation for all of our discussions in this H_Book, except, because we are ignoring the effects of self gravity, <math>~\nabla\Phi</math> is set to zero in the Euler equation and we drop the Poisson equation altogether.  Specifically, the relevant set of governing equations is, the


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<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />


{{User:Tohline/Math/EQ_FirstLaw02}} ,
{{User:Tohline/Math/EQ_FirstLaw02}} .
</div>
</div>


supplemented by an ideal gas equation of state, specifically,
We supplement this set of equations with an ideal gas equation of state, specifically,
<div align="center">
<div align="center">
{{User:Tohline/Math/EQ_EOSideal02}}.
{{User:Tohline/Math/EQ_EOSideal02}} ,
</div>
</div>
As a result, the adiabatic form of the <math>1^\mathrm{st}</math> law of thermodynamics can be written as,
in which case the adiabatic form of the <math>1^\mathrm{st}</math> law of thermodynamics may be written as,
<div align="center">
<div align="center">
<math>
<math>
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===Perturbation then Linearization of Equations===
===Perturbation then Linearization of Equations===


Following Landau and Lifshitz (1975) &#8212; text in green is taken ''verbatum'' from Chapter VIII (pp. 245-248) of LL75 &#8212; we <FONT COLOR="#007700">begin by investigating small oscillations; an oscillatory motion of small amplitude in a compressible fluid is called a ''sound wave.''</FONT>  <FONT COLOR="#007700">Since the oscillations are small, the velocity</FONT> {{User:Tohline/Math/VAR_VelocityVector01}} <FONT COLOR="#007700">is small also, so that the term</FONT>,
Following Landau and Lifshitz (1975) &#8212; text in green is taken ''verbatum'' from Chapter VIII (pp. 245-248) of LL75 &#8212; we <FONT COLOR="#007700">begin by investigating small oscillations; an oscillatory motion of small amplitude in a compressible fluid is called a ''sound wave.''</FONT> Given that <FONT COLOR="#007700">the relative changes in the fluid density and pressure are small</FONT>, in this ''Eulerian'' analysis where we are investigating how conditions vary with time at a fixed point in space, <math>~\vec{r}</math>, <FONT COLOR="#007700">we can write the variables {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}} in the form</FONT>,
<div align="center">
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~P_0 + P_1(\vec{r},t)  \, ,</math>
  </td>
</tr>
 
<tr>
  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0 + \rho_1(\vec{r},t) \, ,</math>
  </td>
</tr>
</table>
</div>
<FONT COLOR="#007700">where <math>~\rho_0</math> and <math>~P_0</math> are the constant</FONT> (both in space and time) <FONT COLOR="#007700">equilibrium density and pressure, and <math>~\rho_1</math> and <math>~P_1</math> are their variations in the sound wave</FONT> <math>~(|\rho_1/\rho_0 | \ll 1, | P_1/P_0 | \ll 1)</math>. <FONT COLOR="#007700">Since the oscillations are small</FONT> &#8212; and because we are assuming that the fluid is initially stationary <math>~(\mathrm{i.e.,}~\vec{v}_0 = 0)</math> &#8212; <FONT COLOR="#007700">the velocity</FONT> {{User:Tohline/Math/VAR_VelocityVector01}} <FONT COLOR="#007700">is small also</FONT>.  In what follows, by definition, <math>~P_1</math>, <math>~\rho_1</math>, and <math>~\vec{v}</math> are considered to be of first order in smallness, while products of these quantities that are of second order in smallness.
 
Substituting the expression for <math>~\rho</math> into the lefthand side of the continuity equation and <FONT COLOR="#007700">neglecting small quantities of the second order</FONT>, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~~\frac{\partial}{\partial t} (\rho_0 + \rho_1) + \nabla\cdot [(\rho_0 + \rho_1)\vec{v}]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\cancelto{0}{\frac{\partial \rho_0}{\partial t}} + \frac{\partial \rho_1}{\partial t} 
+ \nabla\cdot (\rho_0 \vec{v}) + \nabla\cdot\cancelto{\mathrm{small}}{(\rho_1\vec{v} )}
</math>
  </td>
</tr>
 
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \rho_1}{\partial t}  + \rho_0\nabla\cdot \vec{v} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, in the first line, the first term on the righthand side has been set to zero because <math>~\rho_0</math> is independent of time and, in the second line, <math>~\rho_0</math> has been pulled outside of the divergence operator because we have assumed that the initial equilibrium state is homogeneous.  Hence, we have (see, also, equation 63.2 of [[User:Tohline/Appendix/References#LL75|LL75]]) the,
<div align="center">
<font color="#770000">'''Linearized Continuity Equation'''</font><br />
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~
\frac{\partial \rho_1}{\partial t}  + \rho_0\nabla\cdot \vec{v}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>
 
Next, we note that <FONT COLOR="#007700">the term,</FONT>
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<div align="center">
<table border="0" cellpadding="5">
<table border="0" cellpadding="5">
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</table>
</table>
</div>
</div>
<FONT COLOR="#007700">in Euler's equation may be neglected. For the same reason the relative changes in the fluid density and pressure are small.  We can write the variables {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}} in the form</FONT>,
<FONT COLOR="#007700">in Euler's equation may be neglected</FONT> because it is of second order in smallness.   Substituting the expressions for <math>~\rho</math> and <math>~P</math> into the righthand side of the Euler equation and <FONT COLOR="#007700">neglecting small quantities of the second order</FONT>, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~\frac{1}{(\rho_0 + \rho_1)} \nabla (P_0 + P_1)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\rho_0} \biggl( 1 + \frac{\rho_1}{\rho_0} \biggr)^{-1} \biggl[ \cancelto{0}{\nabla P_0} + \nabla P_1\biggr]
</math>
  </td>
</tr>
 
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\rho_0} \biggl[ 1 - \frac{\rho_1}{\rho_0} + \cancelto{\mathrm{small}}{\biggl(\frac{\rho_1}{\rho_0} \biggr)^2} + \cdots \biggr] \nabla P_1
</math>
  </td>
</tr>
 
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\rho_0} \nabla P_1 - \frac{1}{\rho_0^2} \cancelto{\mathrm{small}}{(\rho_1 \nabla P_1)} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, in the first line, <math>~\nabla P_0</math> has been set to zero because we have assumed that the initial equilibrium state is homogeneous, and the binomial theorem has been used to obtain the expression on the righthand side of the second line.


=See Also=
=See Also=
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* [http://en.wikipedia.org/wiki/Wave_equation Wave Equation]
* [http://en.wikipedia.org/wiki/Wave_equation Wave Equation]
* [http://www.astronomy.ohio-state.edu/~dhw/A825/notes6.pdf Sound Waves and Gravitational Instability] &#8212; class notes provided online by David H. Weinberg (The Ohio State University)
* [http://www.astronomy.ohio-state.edu/~dhw/A825/notes6.pdf Sound Waves and Gravitational Instability] &#8212; class notes provided online by David H. Weinberg (The Ohio State University)


{{LSU_HBook_footer}}
{{LSU_HBook_footer}}

Revision as of 22:30, 7 December 2014

Sound Waves

Whitworth's (1981) Isothermal Free-Energy Surface
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A standard technique that is used throughout astrophysics to test the stability of self-gravitating fluids involves perturbing physical variables away from their initial (usually equilibrium) values then linearizing each of the principal governing equations before seeking time-dependent behavior of the variables that simultaneously satisfies all of the equations. When the effects of the fluid's self gravity are ignored and this analysis technique is applied to an initially homogeneous medium, the combined set of linearized governing equations generates a wave equation — whose general properties are well documented throughout the mathematics and physical sciences literature — that, specifically in our case, governs the propagation of sound waves. It is quite advantageous, therefore, to examine how the wave equation is derived in the context of an analysis of sound waves before applying the standard perturbation & linearization technique to inhomogeneous and self-gravitating fluids.

In what follows, we borrow heavily from Chapter VIII of Landau & Lifshitz (1975), as it provides an excellent introductory discussion of sound waves.

Assembling the Key Relations

Governing Equations and Supplemental Relations

We begin with the set of principal governing equations that provides the foundation for all of our discussions in this H_Book, except, because we are ignoring the effects of self gravity, <math>~\nabla\Phi</math> is set to zero in the Euler equation and we drop the Poisson equation altogether. Specifically, the relevant set of governing equations is, the

Eulerian Representation
of the Continuity Equation,

<math>~\frac{\partial\rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0</math>


Eulerian Representation
of the Euler Equation,

<math>\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla) \vec{v}= - \frac{1}{\rho} \nabla P </math>


Adiabatic Form of the
First Law of Thermodynamics

<math>~\frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr) = 0</math> .

We supplement this set of equations with an ideal gas equation of state, specifically,

<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math> ,

in which case the adiabatic form of the <math>1^\mathrm{st}</math> law of thermodynamics may be written as,

<math> \rho \frac{dP}{dt} - \gamma_\mathrm{g} P \frac{d\rho}{dt} = 0 </math>

<math> \Rightarrow ~~~ \frac{d\ln P}{d\ln\rho} = \gamma_\mathrm{g} \, . </math>

Perturbation then Linearization of Equations

Following Landau and Lifshitz (1975) — text in green is taken verbatum from Chapter VIII (pp. 245-248) of LL75 — we begin by investigating small oscillations; an oscillatory motion of small amplitude in a compressible fluid is called a sound wave. Given that the relative changes in the fluid density and pressure are small, in this Eulerian analysis where we are investigating how conditions vary with time at a fixed point in space, <math>~\vec{r}</math>, we can write the variables <math>~P</math> and <math>~\rho</math> in the form,

<math>~P</math>

<math>~=</math>

<math>~P_0 + P_1(\vec{r},t) \, ,</math>

<math>~\rho</math>

<math>~=</math>

<math>~\rho_0 + \rho_1(\vec{r},t) \, ,</math>

where <math>~\rho_0</math> and <math>~P_0</math> are the constant (both in space and time) equilibrium density and pressure, and <math>~\rho_1</math> and <math>~P_1</math> are their variations in the sound wave <math>~(|\rho_1/\rho_0 | \ll 1, | P_1/P_0 | \ll 1)</math>. Since the oscillations are small — and because we are assuming that the fluid is initially stationary <math>~(\mathrm{i.e.,}~\vec{v}_0 = 0)</math> — the velocity <math>~\vec{v}</math> is small also. In what follows, by definition, <math>~P_1</math>, <math>~\rho_1</math>, and <math>~\vec{v}</math> are considered to be of first order in smallness, while products of these quantities that are of second order in smallness.

Substituting the expression for <math>~\rho</math> into the lefthand side of the continuity equation and neglecting small quantities of the second order, we have,

<math>~~\frac{\partial}{\partial t} (\rho_0 + \rho_1) + \nabla\cdot [(\rho_0 + \rho_1)\vec{v}]</math>

<math>~=</math>

<math>~ \cancelto{0}{\frac{\partial \rho_0}{\partial t}} + \frac{\partial \rho_1}{\partial t} + \nabla\cdot (\rho_0 \vec{v}) + \nabla\cdot\cancelto{\mathrm{small}}{(\rho_1\vec{v} )} </math>

 

<math>~\approx</math>

<math>~ \frac{\partial \rho_1}{\partial t} + \rho_0\nabla\cdot \vec{v} \, , </math>

where, in the first line, the first term on the righthand side has been set to zero because <math>~\rho_0</math> is independent of time and, in the second line, <math>~\rho_0</math> has been pulled outside of the divergence operator because we have assumed that the initial equilibrium state is homogeneous. Hence, we have (see, also, equation 63.2 of LL75) the,

Linearized Continuity Equation

<math>~ \frac{\partial \rho_1}{\partial t} + \rho_0\nabla\cdot \vec{v} </math>

<math>~=</math>

<math>~0 \, .</math>

Next, we note that the term,

<math>(\vec{v} \cdot \nabla)\vec{v} \, ,</math>

in Euler's equation may be neglected because it is of second order in smallness. Substituting the expressions for <math>~\rho</math> and <math>~P</math> into the righthand side of the Euler equation and neglecting small quantities of the second order, we have,

<math>~\frac{1}{(\rho_0 + \rho_1)} \nabla (P_0 + P_1)</math>

<math>~=</math>

<math>~ \frac{1}{\rho_0} \biggl( 1 + \frac{\rho_1}{\rho_0} \biggr)^{-1} \biggl[ \cancelto{0}{\nabla P_0} + \nabla P_1\biggr] </math>

 

<math>~=</math>

<math>~ \frac{1}{\rho_0} \biggl[ 1 - \frac{\rho_1}{\rho_0} + \cancelto{\mathrm{small}}{\biggl(\frac{\rho_1}{\rho_0} \biggr)^2} + \cdots \biggr] \nabla P_1 </math>

 

<math>~=</math>

<math>~ \frac{1}{\rho_0} \nabla P_1 - \frac{1}{\rho_0^2} \cancelto{\mathrm{small}}{(\rho_1 \nabla P_1)} \, , </math>

where, in the first line, <math>~\nabla P_0</math> has been set to zero because we have assumed that the initial equilibrium state is homogeneous, and the binomial theorem has been used to obtain the expression on the righthand side of the second line.

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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